Optical metrology techniques generally referred to as scatterometry offer the potential to characterize parameters of a workpiece during a manufacturing process. In practice, light is directed onto a periodic grating formed in a workpiece and a spectrum of reflected light is measured and analyzed to characterize the grating parameters. Characterization parameters may include critical dimensions (CD), sidewall angle (SWA), feature height (HT) and any which vary a material's reflectivity and refractive index. Characterization of the grating may thereby characterize the workpiece as well as manufacturing process employed in the formation of the grating and the workpiece.
Analysis of a measured spectrum typically involves comparing the measurement data to theoretical spectra in a library to deduce the parameters that best describe the measured grating. A theoretical spectrum for a set of grating parameter values can be computed using rigorous diffraction modeling algorithms, such as Rigorous Coupled Wave Analysis (RCWA). However, computing the reflection coefficient of scattered light, as a function of wavelength, from a periodic grating of a given parameterized geometry can be very slow when solving the inverse grating diffraction problem, whereby analysis of the diffracted light measurements via regression returns the estimated parameter as a function of the measured spectra. Thus, a method of estimating grating parameters more rapidly and with sufficient accuracy is needed.
Generally, a neural network may function as multidimensional Lagrange interpolator. Given a set of vector valued inputs, a corresponding set of values of a function to be interpolated, a suitable neural network topology and sufficient training of the network, the output of the neural network can be approximately equal to an original function evaluation. FIG. 1 depicts the topology of a conventional simple neural network (SNN) that is referred to as a one hidden layer network (one input p, one hidden layer, one output layer). Neural network 100 has an input vector, p, layer weight matrices, Wi, layer bias vectors, di, nonlinear basis function, s(u)=[σ(u)1 σ(u)2 . . . σ(u)2]T, and output weight vector, v to provide output vector y.